Bottom Line:
Based on our analysis, we draw several conclusions for the use of those internal measures on microarray data.Another important novel conclusion that can be drawn from our analysis is that all the measures we have considered show severe limitations on large datasets, either due to computational demand (Consensus, as already mentioned, Clest and Gap) or to lack of precision (all of the other measures, including their approximations).The software and datasets are available under the GNU GPL on the supplementary material web page.

Background: Inferring cluster structure in microarray datasets is a fundamental task for the so-called -omic sciences. It is also a fundamental question in Statistics, Data Analysis and Classification, in particular with regard to the prediction of the number of clusters in a dataset, usually established via internal validation measures. Despite the wealth of internal measures available in the literature, new ones have been recently proposed, some of them specifically for microarray data.

Results: We consider five such measures: Clest, Consensus (Consensus Clustering), FOM (Figure of Merit), Gap (Gap Statistics) and ME (Model Explorer), in addition to the classic WCSS (Within Cluster Sum-of-Squares) and KL (Krzanowski and Lai index). We perform extensive experiments on six benchmark microarray datasets, using both Hierarchical and K-means clustering algorithms, and we provide an analysis assessing both the intrinsic ability of a measure to predict the correct number of clusters in a dataset and its merit relative to the other measures. We pay particular attention both to precision and speed. Moreover, we also provide various fast approximation algorithms for the computation of Gap, FOM and WCSS. The main result is a hierarchy of those measures in terms of precision and speed, highlighting some of their merits and limitations not reported before in the literature.

Conclusion: Based on our analysis, we draw several conclusions for the use of those internal measures on microarray data. We report the main ones. Consensus is by far the best performer in terms of predictive power and remarkably algorithm-independent. Unfortunately, on large datasets, it may be of no use because of its non-trivial computer time demand (weeks on a state of the art PC). FOM is the second best performer although, quite surprisingly, it may not be competitive in this scenario: it has essentially the same predictive power of WCSS but it is from 6 to 100 times slower in time, depending on the dataset. The approximation algorithms for the computation of FOM, Gap and WCSS perform very well, i.e., they are faster while still granting a very close approximation of FOM and WCSS. The approximation algorithm for the computation of Gap deserves to be singled-out since it has a predictive power far better than Gap, it is competitive with the other measures, but it is at least two order of magnitude faster in time with respect to Gap. Another important novel conclusion that can be drawn from our analysis is that all the measures we have considered show severe limitations on large datasets, either due to computational demand (Consensus, as already mentioned, Clest and Gap) or to lack of precision (all of the other measures, including their approximations). The software and datasets are available under the GNU GPL on the supplementary material web page.

Figure 5: An example of number of cluster prediction with the use of ME. Consider the computation of the ME algorithm on the dataset of Fig. 2(a). The histograms plotting the SIM values distribution for increasing values of k are shown above. The transition allowing for the prediction of k* takes place at k = 2 for a correct prediction of k* = 2.

Mentions:
Once that the SIM array is computed, its values are histogrammed, separately for each value of k, i.e., by rows. The optimal number of clusters is predicted to be the lowest value of k such that there is a transition of the SIM value distribution from being close to one to a wider range of values. An example is given in Fig. 5, where the transition described above takes place at k = 2 for a correct prediction of k* = 2.

Figure 5: An example of number of cluster prediction with the use of ME. Consider the computation of the ME algorithm on the dataset of Fig. 2(a). The histograms plotting the SIM values distribution for increasing values of k are shown above. The transition allowing for the prediction of k* takes place at k = 2 for a correct prediction of k* = 2.

Mentions:
Once that the SIM array is computed, its values are histogrammed, separately for each value of k, i.e., by rows. The optimal number of clusters is predicted to be the lowest value of k such that there is a transition of the SIM value distribution from being close to one to a wider range of values. An example is given in Fig. 5, where the transition described above takes place at k = 2 for a correct prediction of k* = 2.

Bottom Line:
Based on our analysis, we draw several conclusions for the use of those internal measures on microarray data.Another important novel conclusion that can be drawn from our analysis is that all the measures we have considered show severe limitations on large datasets, either due to computational demand (Consensus, as already mentioned, Clest and Gap) or to lack of precision (all of the other measures, including their approximations).The software and datasets are available under the GNU GPL on the supplementary material web page.

Background: Inferring cluster structure in microarray datasets is a fundamental task for the so-called -omic sciences. It is also a fundamental question in Statistics, Data Analysis and Classification, in particular with regard to the prediction of the number of clusters in a dataset, usually established via internal validation measures. Despite the wealth of internal measures available in the literature, new ones have been recently proposed, some of them specifically for microarray data.

Results: We consider five such measures: Clest, Consensus (Consensus Clustering), FOM (Figure of Merit), Gap (Gap Statistics) and ME (Model Explorer), in addition to the classic WCSS (Within Cluster Sum-of-Squares) and KL (Krzanowski and Lai index). We perform extensive experiments on six benchmark microarray datasets, using both Hierarchical and K-means clustering algorithms, and we provide an analysis assessing both the intrinsic ability of a measure to predict the correct number of clusters in a dataset and its merit relative to the other measures. We pay particular attention both to precision and speed. Moreover, we also provide various fast approximation algorithms for the computation of Gap, FOM and WCSS. The main result is a hierarchy of those measures in terms of precision and speed, highlighting some of their merits and limitations not reported before in the literature.

Conclusion: Based on our analysis, we draw several conclusions for the use of those internal measures on microarray data. We report the main ones. Consensus is by far the best performer in terms of predictive power and remarkably algorithm-independent. Unfortunately, on large datasets, it may be of no use because of its non-trivial computer time demand (weeks on a state of the art PC). FOM is the second best performer although, quite surprisingly, it may not be competitive in this scenario: it has essentially the same predictive power of WCSS but it is from 6 to 100 times slower in time, depending on the dataset. The approximation algorithms for the computation of FOM, Gap and WCSS perform very well, i.e., they are faster while still granting a very close approximation of FOM and WCSS. The approximation algorithm for the computation of Gap deserves to be singled-out since it has a predictive power far better than Gap, it is competitive with the other measures, but it is at least two order of magnitude faster in time with respect to Gap. Another important novel conclusion that can be drawn from our analysis is that all the measures we have considered show severe limitations on large datasets, either due to computational demand (Consensus, as already mentioned, Clest and Gap) or to lack of precision (all of the other measures, including their approximations). The software and datasets are available under the GNU GPL on the supplementary material web page.